Some systems of nonlinear wave equations admit global solutions for all sufficiently small initial data, while others do not. The (classical) null condition guarantees that such a result holds, but it is too strong to capture certain systems -- most famously the Einstein equations -- which nevertheless admit global solutions for small initial data. The weak null condition has been proposed as a sufficient condition for such a result to hold; it takes the form of a condition on a related set of nonlinear ODEs known as the "asymptotic system". Previous results in this direction have required certain structural conditions on the asymptotic system in addition to the weak null condition. In this work we show that, if the solutions to the asymptotic system are bounded (given small initial data), and, in addition, if these solutions are stable against rapidly decaying perturbations, then the corresponding system of nonlinear wave equations admits global solutions for all sufficiently small initial data. This avoids any direct assumptions on the structure of the nonlinear terms. We also give an example of a class of systems obeying this condition but not obeying previously identified structural conditions. For this class the asymptotic system arises as a generalisation of the "Euler equations" for rigid body motion, associated with a left-invariant Hamiltonian flow on a finite dimensional Lie group. This work relies heavily on the author's previous work, "The weak null condition and global existence using the p-weighted energy method".
We consider positive solutions of a fractional Lane-Emden type problem in a bounded domain with Dirichlet conditions. We show that uniqueness and nondegeneracy hold for the asymptotically linear problem in general domains. Furthermore, we also prove that all the known uniqueness and nondegeneracy results in the local case extend to the nonlocal regime when the fractional parameter s is sufficiently close to 1.
Global existence and boundedness of classical solutions are shown for a parabolic-elliptic chemotaxis system with local sensing when the motility function is assumed to be unbounded at infinity. The cornerstone of the proof is the derivation of $L^\infty$-estimates on the second component of the system and is achieved by various comparison arguments.
In this paper we consider the incompressible Euler equation in a simply-connected bounded planar domain. We study the confinement of the vorticity around a stationary point vortex. We show that the power law confinement around the center of the unit disk obtained in [2] remains true in the case of a stationary point vortex in a simply-connected bounded domain. The domain and the stationary point vortex must satisfy a condition expressed in terms of the conformal mapping from the domain to the unit disk. Explicit examples are discussed at the end.
A 7-dimensional area-minimizing embedded hypersurface $M$ will in general have a discrete singular set. The same is true if $M$ is stable, or has bounded index, provided $H^6(sing M) = 0$. We show that if $M_i$ are a sequence of such minimal hypersurfaces which are minimizing, stable, or have bounded index, then $M_i$ can limit to a singular $M$ with only very controlled geometry, topology, and singular set. We show one can always "parameterize" a subsequence $i'$ with controlled bi-Lipschitz maps $ϕ_{i'}$ taking $ϕ_{i'}(M_{1'}) = M_{i'}$. As a consequence, we prove the space of smooth, closed…
We prove the optimal global regularity of nonnegative solutions to the porous medium equation in smooth bounded domains with the zero Dirichlet boundary condition after certain waiting time $T^*$. More precisely, we show that solutions are $C^{2,α}(\overlineΩ)$ in space, with $α=\frac1m$, and $C^\infty$ in time (uniformly in $x\in \overlineΩ$), for $t>T^*$. Furthermore, this allows us to refine the asymptotics of solutions for large times, improving the best known results so far in two ways: we establish a faster rate of convergence $O(t^{-1-γ})$, and we prove that the convergence holds in the…
The global boundedness and asymptotic behavior are investigate for the solution of time-space fractional non-local reaction-diffusion equation (TSFNRDE) $$ \frac{\partial^{α}u}{\partial t^{α}}=-(-Δ)^{s} u+μu^{2}(1-kJ*u)-γu, \qquad(x,t)\in\mathbb{R}^{N}\times(0,+\infty),$$ where $s\in(0,1),α\in(0,1), N \leq 2$. The operator $\partial_{t}^{α}$ is the Caputo fractional derivative, which $-(-Δ)^{s}$ is the fractional Laplacian operator. For appropriate assumptions on $J$, it is proved that for homogeneous Dirichlet boundary condition, this problem admits a global bounded weak solution for $N=1$, w…
In this paper we show that there exists a family of domains $Ω_{\varepsilon}\subseteq\mathbb{R}^N$ with $N\ge2$, such that the $stable$ solution of the problem \[ \begin{cases} -Δu= g(u)&\hbox{in }Ω_\varepsilon\\ u>0&\hbox{in }Ω_\varepsilon\\ u=0&\hbox{on }\partialΩ_\varepsilon \end{cases} \] admits $k$ critical points with $k\ge2$. Moreover the sets $Ω_\varepsilon's$ are star-shaped and "close" to a strip as $\varepsilon\to0$. Next, if $g(u)\equiv1$ and $N\ge3$ we exhibit a family of domain $Ω_\varepsilon's$ with $positive$ $mean$ $curvature$ and solutions $u_\varepsilon $ which have $k$ crit…
We study an asymptotic analysis of a coupled system of kinetic and fluid equations. More precisely, we deal with the nonlinear Vlasov-Fokker-Planck equation coupled with the compressible isentropic Navier-Stokes system through a drag force in a bounded domain with the specular reflection boundary condition for the kinetic equation and homogeneous Dirichlet boundary condition for the fluid system. We establish a rigorous hydrodynamic limit corresponding to strong noise and local alignment force. The limiting system is a type of two-phase fluid model consisting of the isothermal Euler system and…
We consider the Dirichlet Laplacian $A_q=-Δ+q$ in a bounded domain $Ω\subset \mathbb{R}^d$, $d \ge 3$, with real-valued perturbation $q \in L^{\max(2 , 3 d / 5)}(Ω)$. We examine the stability issue in the inverse problem of determining the electric potential $q$ from the asymptotic behavior of the eigenvalues of $A_q$. Assuming that the boundary measurement of the normal derivative of the eigenfunctions is a square summable sequence in $L^2(\partial Ω)$, we prove that $q$ can be Hölder stably retrieved through knowledge of the asymptotics of the eigenvalues
In the present work, we establish the approximation of nonlinear stochastic partial differential equation (SPDE) driven by cylindrical α-stable Lévy processes via modulation or amplitude equations. We study SPDEs with a cubic nonlinearity, where the deterministic equation is close to a change of stability of the trivial solution. The natural separation of time-scales close to this bifurcation allows us to obtain an amplitude equation describing the essential dynamics of the bifurcating pattern, thus reducing the original infinite dimensional dynamics to a simpler finite-dimensional effective d…
Consider 3D Boltzmann equation in convex domains with diffusive-reflection boundary condition. We study the hydrodynamic limits as the Knudsen number and Strouhal number $ε\rightarrow 0^+$. Using the Hilbert expansion, we rigorously justify that the solution of stationary/evolutionary problem converges to that of the steady/unsteady Navier-Stokes-Fourier system. This is the first paper to justify the hydrodynamic limits of nonlinear Boltzmann equations with hard-sphere collision kernel in bounded domain in the $L^{\infty}$ sense. The proof relies on a novel analysis on the boundary layer effec…
